Chemical potentials of polymers

Conditions of phase equilibrium require that the chemical potential of polymer in each phase and that of solvent in each phase be equal ... 

Polymerization reactions are in most cases conducted under conditions remote from polymer-monomer equilibrium, i.e., these reactions are, to a great extent, irreversible. A polymer can have different molecular and supramolecular structures for example it can be iso- or syndiotactic, amorphous, or crystalline. Differences in the chemical potential of polymers with different structures are small in comparison with the changes observed in the chemical potential at conversion of a monomer into a polymer. This means that the possibility for a polymer of a certain structure to be formed will be determined by kinetic causes the nature of the catalyst, solvent, etc. According to the scheme in Fig. 5, the polymer with structure 2 will be mainly produced. 

What is then the basis for the thermodynamic approach One can select the reaction conditions in such a way that variations in the chemical potential during the process will be comparable to the difference in the chemical potentials of polymers... 

Kammer [1977] considered the interface between two polymers from the basic thermodynamic point of view. He derived a simple relation Vj = A 0g/S, where A 0g is the excess chemical potential of polymer B in the mixture, and S is the molar area of the interface. Near the spinodal decomposition, using Cahn-Hilliard gradient theory, he calculated ... 

If in the system P+LMWI< at some temperature 1 the chemical potential of mixing of the polymer in solution per mole of monomeric units A//2 at some concentration I i = rj proves to be equal to the difference of the chemical potentials of polymer in the crystal and amorphous phases then there will be two phases in equilibrium, namely, polymer... 

The roots of this method can be traced back to the pioneoing work of the Rosenbluths in the 1950s [64]. However, the CCB method in reality is a direct descendant of the Scanning method of Meirovich [65-68], in partkular of the version for attractive random walks [68]. A related idea was introduced by Harris and Rice [69]. The method has recently attracted much intnest, and has been fully developed as a simulation tool through the work of Siepmann [42], Frenkel et al. [43], and Siepmann and Frenkel [44]. de Pablo et al. [45] implemented the CCB method for the off-lattice treatment of realistic polymer systems. The initial off-lattice applications have demonstrated that the method can be used in a wide variety of important problems in polymer systems, most notably the determination of equilibrium thermodynamic properties, chemical potentials of polymers, soluUlitks d gi t mol ades in polymer melts, studies of phase transitions, and polymer-sdivent interactions in supercritical fluids [70-72]. 

At the equilibrium melting point, the chemical potential of polymers in the amorphous phase is equal to that in the crystalline phase jf, thus... 

The computational problem of polymer phase equilibrium is to provide an adequate representation of the chemical potentials of each component in solution as a function of temperature, pressure, and composition. 

Another general type of behavior that occurs in polymer manufacture is shown in Figure 3. In many polymer processing operations, it is necessary to remove one or more solvents from the concentrated polymer at moderately low pressures. In such an instance, the phase equilibrium computation can be carried out if the chemical potential of the solvent in the polymer phase can be computed. Conditions of phase equilibrium require that the chemical potential of the solvent in the vapor phase be equal to that of the solvent in the liquid (polymer) phase. Note that the polymer is essentially involatile and is not present in the vapor phase. 

There is a similar expression for polymer activity. However, if the fluid being sorbed by the polymer is a supercritical gas, it is most useful to use chemical potential for phase equilibrium calculations rather than activity. For example, at equilibrium between the fluid phase (gas) and polymer phase, the chemical potential of the gas in the fluid phase is equal to that in the liquid phase. An expression for the equality of chemical potentials is given by Cheng (12). 

Differentiation of Eq. (22) with respect to ri2 yields for the chemical potential of the polymeric solute relative to the pure liquid polymer as standard state... 

The chemical potential of the species of size x in a heterogeneous polymer, obtained by differentiating Eq. (23) with respect to is... 

In a similar way the chemical potential of the polymer unit readily reduces for large x to... 

Fig. 120.—The chemical potential of the solvent in a binary solution containing polymer at low concentrations vi). Curves have been calculated according to Eq. (XII-26) for a = 1000 and the values of dicated with each curve. ...

Donnan-Type Equilibria in Polyelectrolyte Gels.—In a somewhat more rigorous fashion we consider the reduction of the chemical potential of the solvent in the swollen gel to be separable into three terms which severally represent the changes due to the mixing of polymer and solvent, to the mixing with the mobile ionic constituents, and to the elastic deformation of the network. Symbolically... 

Vapour pressure osmometry is the second experimental technique based on colligative properties with importance for molar mass determination. The vapour pressure of the solvent above a (polymer) solution is determined by the requirement that the chemical potential of the solvent in the vapour and in the liquid phase must be identical. For ideal solutions the change of the vapour pressure p of the solvent due to the presence of the solute with molar volume V/1 is given by... 

One of the interests in confined polymers arises from adsorption behavior— that is, the intake or partitioning of polymers into porous media. Simulation of confined polymers in equilibrium with a bulk fluid requires simulations where the chemical potentials of the bulk and confined polymers are equal. This is a difficult task because simulations of polymers at constant chemical potential require the insertion of molecules into the fluid, which has poor statistics for long chains. Several methods for simulating polymers at constant chemical potential have been proposed. These include biased insertion methods [61,62], novel simulation ensembles [63,64], and simulations where the pore is physically connected to a large bulk reservoir [42]. Although these methods are promising, so far they have not been implemented in an extensive study of the partitioning of polymers into porous media. This is a fruitful avenue for future research. 

In Eq. (2), mi is the chemical potential of the solvent in the polymer gel and /al 0 is the chemical potential of the pure solvent. At equilibrium, the difference between the chemical potentials of the solvent outside and inside the gel must be zero. Therefore, changes of the chemical potential due to mixing and elastic forces must balance each other. The change of chemical potential due to mixing can be expressed using heat and entropy of mixing. 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

It is fortunate that theory has been extended to take into account selective interactions in multicomponent systems, and it is seen from Eq. (91) (which is the expression used for the plots in Fig. 42 b) that the intercept at infinite dilution of protein or other solute does give the reciprocal of its correct molecular weight M2. This procedure is a straightforward one whereby one specifies within the constant K [Eq. (24)] a specific refractive index increment (9n7dc2)TiM. The subscript (i (a shorter way of writing subscripts jUj and ju3) signifies that the increments are to be taken at constant chemical potential of all diffusible solutes, that is, the components other than the polymer. This constitutes the osmotic pressure condition whereby only the macromolecule (component-2) is non-diffusible through a semi-permeable membrane. The quantity... 

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